66 ANALYTICAL MKCHANICS The curve, Fig. 43, defined by equation (15) is called an exponential curve. It has an interesting property, namely, its ordinate is doubled every time a constant value P is added to its abscissa. This constant is called the half-value period of the curve. The value of P may be determined in the following manner. By the definition of P and from equation (15) we have Fi Dividing equation (16) by equation (15) we get or P = a log,, 2. LENGTH OF CABLE.—In order to find the length in terms of the span eliminate y between equations (7) and (11). This gives / x _;r\ Q _ :_ I pa __p a f I 7 \ o •— \ t/ ^^ \j I \ I i / w rt \ / \ * * / = X+J 5! + 1 *+ ... f (IK) £t • o a & * o * *i• * o u where the right member of equation (IS) is obtained by expanding the right-hand member of equation (17) by Maclaurin's Theorem. If D and L denote the span and the length of the rubles respectively, we have s = f L when or«i/>. Therefore substituting these values of s and x in equation (18) awl replacing a by its value we obtain 48 (19)